We describe the weight functions for which Hardy's inequality of nonincreasing functions is satisfied. Further we characterize the pairs of weight functions for which the Laplace transform , with monotone function , is bounded from the weighted Lebesgue space to .

In this paper, we study property () and B-convexity in reflexive Banach spaces. It is shown that k-uniform convexity implies B-convexity and property (). We also show that there is a Banach space with property () which cannot be equivalently renormed to be B-convex.

In this paper, we define and study the C-integral and the strong C-integral of functions mapping an interval [a,b] into a Banach space X. We prove that the C-integral and the strong C-integral are equivalent if and only if the Banach space is finite dimensional, We also consider the property of primitives corresponding to Banach-valued functions with respect to the C-integral and the strong C-integral.

In this paper, using the Black-Scholes analysis, we will derive the partial differential equation of convertible bonds with both non-stochastic and stochastic interest rate. We also find numerical solutions of convertible bonds equation with known interest rate using the finite element method.

In this note the notion of interval-valued fuzzy BG-algebras (briefly, i-v fuzzy BG-algebras), the level and strong level BG-subalgebra is introduced. Then we state and prove some theorems which determine the relationship between these notions and BG-subalgebras. The images and inverse images of i-v fuzzy BG-subalgebras are defined, and how the homomorphic images and inverse images of i-v fuzzy BG-subalgebra becomes i-v fuzzy BG-algebras are studied.

In this paper, we investigate the mutual relation among short exact sequences of amalgamated free products which involve augmentation ideals and relation modules. In particular, we find out commutative diagrams having a steady structure in the sense that all of their three columns and rows are short exact sequences.

We study some dynamical properties in the context of bitransformation groups, and show that if (H,X,T) is a bitransformation group such that (H,X) is almost periodic and (X/H,T) is pointwise almost periodic and , then is a compact topological group and when H is abelian, where E(H,X) is the enveloping semigroup of the transformation group (H,X).

In this paper we introduce the concepts of several types of quasi-smooth -compactness in terms of the concepts of weak smooth -closure and weak smooth -interior of a fuzzy set in smooth topological spaces and investigate some of their properties.

The weakened forms of the ()-continuous function are introduced and their basic properties are investigated in concern with the other weakened continuous function. The open property of a function and the extremal disconnectedness of the spaces are crucial tools for the survey of these functions.

We consider a semilinear elliptic boundary value problem with Dirichlet boundary condition in and is the eigenfuction corresponding to . We have a concern with the multiplicity of solutions of the equation when < a < < b < .

Lipschitzian semigroup is a semigroup of Lipschitz operators which contains semigroup and nonlinear semigroup. In this paper, we establish the cannonical exponential formula of Lipschitzian semigroup from its Lie generator and the approximation theorem by Laplace transform approach to Lipschitzian semigroup.

We introduce the resistant length and examine its properties. We also consider the geometric applications of resistant length to the boundary behavior of analytic functions, conformal mappings and derive the theorem in connection with the cluster sets, purely geometric problems. The method of resistant length leads a simple proofs of theorems. So it shows us the usefulness of the method of resistant length.

In this paper, we give sufficient conditions for a.s. convergence of weighted sums of integrable fuzzy random variables. As a result, we obtain strong laws of large numbers for weighted sums of independent fuzzy random variables.